A quantum Monte Carlo study of high pressure solid and liquid hydrogen

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Department of Physics

Graduate School in Astronomical, Chemical, Mathematical, Physical and Earth

Sciences “Vito Volterra” - XXIX cycle

Doctoral Thesis

A Quantum Monte Carlo study of high pressure

solid and liquid hydrogen

Giovanni Rillo

Advisors:

Prof. Andrea Pelissetto Prof. Carlo Pierleoni

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Introduction iii

1 Hydrogen at high pressures 1

1.1 Experimental solid phase diagram . . . 2

1.1.1 Phase I . . . 3

1.1.2 The broken symmetry phases: phase II and III . . . 3

1.1.3 Mixed phases: phase IV and IV’ . . . 5

1.1.4 Possible phases at higher pressures . . . 5

1.2 Liquid-liquid phase transition . . . 6

2 Simulation methods 8 2.1 The electronic problem and the Born-Oppenheimer approximation . . . 8

2.2 Density Functional Theory . . . 12

2.2.1 The Hohenberg-Kohn theorem . . . 12

2.2.2 The Kohn-Sham equations . . . 15

2.2.3 Practical implementation . . . 17

2.3 Quantum Monte Carlo . . . 19

2.3.1 Monte Carlo methods and the Metropolis algorithm . . . 20

2.3.2 Variational Monte Carlo . . . 22

2.3.3 Our trial wavefunction . . . 23

2.3.3.1 The Kato cusp conditions . . . 23

2.3.3.2 Backflow transformation . . . 26

2.3.4 Twist averaged boundary conditions . . . 27

2.4 Path integrals . . . 28

2.4.1 The primitive approximation . . . 29

2.4.2 The pair product action . . . 31

2.5 Path Integral Molecular Dynamics . . . 32

2.5.1 Path Integral and Langevin Equations . . . 34

2.6 Coupled electron-ion Monte Carlo . . . 36

2.6.1 The Penalty method . . . 37

2.6.2 Technical details . . . 39

2.6.2.1 Trial wavefunction: the orbitals . . . 39

2.6.2.2 Trial wavefunction: the optimization of the coefficients . . . 40

2.6.2.3 Path integrals: proposing the protonic move . . . 41

2.6.2.4 Evaluating the electronic energy differences . . . 42

3 High pressure solid hydrogen 44 3.1 Crystal structures . . . 45

3.1.1 Mixed structures . . . 46

3.2 Previous works . . . 46

3.2.1 Dynamical simulations . . . 49

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3.2.1.1 Classical protons with the PBE functional . . . 49

3.2.1.2 Introducing nuclear quantum effects and comparing PBE with other functionals . . . 50

3.2.2 Static simulations . . . 51

3.3 Setup of the simulations . . . 53

3.3.1 DFT-PIMD . . . 54

3.3.2 CEIMC . . . 54

3.3.2.1 Variational Monte Carlo setup . . . 54

3.3.2.2 Path Integrals setup . . . 55

3.4 Results . . . 56 3.4.1 C2c, T=200 K . . . 57 3.4.2 Cmca12 at T=200 K . . . 63 3.4.3 Electronic properties . . . 64 3.4.4 Pc48 at T=414 K . . . 66 3.4.5 Discussion . . . 69 3.5 Conclusions . . . 71

4 Liquid liquid phase transition 72 4.1 Previous works . . . 72

4.2 CEIMC simulations . . . 75

4.2.1 One body density matrix . . . 75

4.3 Methods . . . 77

4.4 Results . . . 78

4.4.1 Conductivity and one body density matrix . . . 78

4.4.2 Optical properties around the liquid-liquid phase transition . . . 82

4.5 Conclusions . . . 85

A Transport and thermoelectrical kinetic coefficients 89 A.1 Irreversible thermodynamics and kinetic coefficients . . . 89

A.2 Quantum linear response theory . . . 91

A.2.1 Conductivity . . . 92

A.2.2 Kinetic coefficients . . . 93

A.3 DFT implementation . . . 94

A.3.1 Static kinetic coefficients . . . 95

A.3.2 Dynamical conductivity and optical properties . . . 96

Conclusions 89

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Hydrogen is the first element of the periodic table. As such, it is often regarded as the simplest one: the non-relativistic hydrogen atom is a problem exactly solved in many textbooks; the hydrogen molecular ion H₂+ and the diatomic molecule H2 are, correspondingly, the first systems to be

considered when more than one nucleus is involved. As a thermodynamic system, its phase diagram at low pressures is quite standard: at room temperature and ambient pressure, hydrogen is a molecular fluid; upon cooling, it becomes a molecular solid; its critical point is T=33 K and P=1.3 Pa [1].

Nevertheless, even such a simple system becomes really interesting when pressure is increased by several orders of magnitude. Speculations about the existence of a metallic solid state at 25 GPa and 0 K temperature started with Wigner and Huntington [2]; later calculations suggested that this state could become a high-temperature superconductor [3]. When experiments achieved the predicted transition temperature, they did not find a metallic state; on the other hand, they found a rich phase diagram, where several different solid phases exist [4, 5]. Nowadays, the quest for solid metallic hydrogen at low temperature is still an on-going activity.

As temperature is increased above ≈ 1000 K, the system enters the liquid phase: it is important to obtain an accurate equation of state at high temperature and high pressure, in order to model the properties of gas giants, such as Jupiter and Saturn, which are mostly made of hydrogen and helium. Metallic hydrogen, which is yet to be seen in the solid state, was experimentally measured in the liquid phase [6].

Performing experiments at such high pressures is complicated; the information obtained is partial. At low temperatures, the boundaries among the different solid phases can be drawn, but most of their structural properties are still an open problem; at high temperatures, characterizing the insulator-metal transition is hard because of large uncertainties and conflicting results.

Ab Initio simulations can be a valuable tool to complement and interpret experimental data; they can also guide experiments with their predictive power. For condensed matter, Density Functional Theory (DFT) is the method of choice to perform Ab Initio simulations at reasonable computational cost. However, their predictive power for high pressure hydrogen is questioned due to several levels of approximation which will be discussed in our work: in particular, the fact that DFT is plagued by an uncontrolled approximation (the exchange-correlation functional approximation) will be elaborated.

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how CEIMC, combining the Path Integral formalism to treat the nuclear degrees of freedom and the Variational Monte Carlo (VMC) method to accurately compute electronic energies in a Born-Oppenheimer framework, can perform finite temperature simulations without suffering from the same kind of uncontrolled approximation which plagues DFT. We will then apply the method to the low temperature, solid phase and to the high-temperature, liquid one. In the first case, finite temperature simulations of different candidate structures for the various solid phases will be performed, comparing CEIMC results with DFT ones. In the second case, the liquid-liquid phase transition will be investigated, drawing attention to the relationship between molecular dissociation and metallization; to do so, the system will be characterized across the transition with the computation of relevant optical properties.

This work is organized as follows. In Chapter 1, a short review of the experimental phase diagram of high pressure hydrogen is presented, discussing the strengths and the limitations of the experimental techniques employed, which information can be extracted and which cannot. In Chapter 2, the theoretical framework necessary to perform DFT and CEIMC simulations is discussed: Density Functional Theory and Variational Monte Carlo are introduced to compute electronic energies. In particular, the form of the trial wavefunction, the key ingredient of the VMC method, is discussed in detail. To account for quantum nuclear effects, the Path Integral formalism is introduced together with efficient ways of sampling the associated probability distribution: Path Integral Molecular Dynamics (PIMD) and Path Integral Monte Carlo (PIMC). In Chapter 3, after a small review of relevant previous theoretical calculations, we present our results about finite temperature simulations of different candidate structures for solid hydrogen. In particular, we performed DFT-PIMD and CEIMC simulations at T=200 K and T=414 K, analyzing the structural and electronic properties of the system and comparing at the same time the results produced by the two methods. Finally, in Chapter 4, we focus on the liquid-liquid phase transition. After reviewing previous literature, the system is characterized through the computation of optical properties across the transition, deducing its metallic or insulating state. The study of optical properties is also expanded in a larger region of the phase diagram.

Papers associated to the thesis work

• Liquid–liquid phase transition in hydrogen by coupled electron–ion Monte Carlo simulations, Pierleoni, Carlo, Morales, Miguel A., Rillo, Giovanni, Ho lzmann, Markus, Ceperley, David M., PNAS, vol. 113 | no. 18 | 4953–4957 (2016).

• Optical properties of liquid hydrogen across the molecular dissociation, G. Rillo, M.A. Morales, D.M. Ceperley and C. Pierleoni, in preparation

• Coupled Electron-Ion Monte Carlo simulation of the crystalline phases III and IV of molecular hydrogen, G. Rillo, M.A. Morales, D.M. Ceperley and C. Pierleoni, in preparation

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Hydrogen at high pressures

Interest in high pressure hydrogen was firstly drawn by Wigner and Huntington [2], who in 1935 speculated about a possible stable metallic phase of solid atomic hydrogen at pressures higher than 25 GPa. At the time, achieving such high pressures for hydrogen was experimentally unfeasible: H2 was compressed at 2 GPa only in 1956 [7]. In the seventies, the development of diamond anvil

cells allowed to generate pressures of hundreds of GPa under static conditions [8] (more precisely, up to 170 GPa at 25◦C). Solid hydrogen was thus obtained by Mao and Bell at a temperature of 25◦C and at a pressure of 50 GPa in 1979 [9]. The material was found to be transparent, a clear indication that the metallic state was yet to be reached, in contrast with the early prediction by Wigner. Since then, diamond anvil cells have been the standard tool to investigate high pressure hydrogen at relatively low temperatures. Technical improvements in DAC experiments pushed the highest reachable pressure further and further: nowadays, pressures above 300 GPa can be obtained consistently. A rich phase diagram with different solid phases was discovered, even if a full characterization of these phases is still missing.

High pressure hydrogen is also interesting at higher temperatures (thousands of kelvins), where the system is in its liquid phase. Knowledge of hydrogen in this regime is relevant for modeling the interiors of planets like Saturn and Jupiter, made up hydrogen (90 %) and helium (9%). This region of the phase diagram can nowadays be probed using DAC [10, 11], as in the solid case, using special heating techniques. A more complete picture of the liquid state can be achieved using dynamic shock compression, a technique that allows one to drive the system to pressures of ≈ 500 GPa and temperatures of ≈ 10000 K [12, 13]. Using this method, liquid metallic hydrogen was detected at 140 GPa and 3000 K [6]. However, large uncertainties usually affect measurements due to the very dynamic nature of the process: while the existence of metallic hydrogen was proven, the transition from an insulating to a metallic liquid is still poorly characterized, with different experiments obtaining conflicting results [10, 14].

In the following sections, we will discuss the experimental phase diagram of high pressure hydrogen, both in the solid and in the liquid region. The experimental evidence will be described, pointing out at the same time missing relevant features that are still object of discussion.

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1.1 Experimental solid phase diagram

When looking for crystal structures of solid phases, the two techniques of choice are X-ray diffraction and neutron scattering. In particular, X-ray diffraction probes the electronic density and, for molec-ular systems, can identify the positions of the molecmolec-ular centers; neutron scattering, being sensitive to the positions of the individual nuclei, gives information about the single atoms. Unfortunately, the hydrogen cross-section is extremely low in both cases: X-ray diffraction was performed up to 180 GPa [15], while neutron scattering up to 60 GPa [16]. At higher pressures, structural information is extracted through Raman and infrared (IR) spectra.

100

200

300

400 Pressure (GPa)

0

400

800 Temperature (K)

Liquid

IV’

Solid H

₂

III

I

II

IV

V

VI

Figure 1.1: Experimental phase diagram of solid hydrogen. Black continuous lines indicate the melting line as well as the transition lines between different experimentally detected [4, 5] crystalline phases: I–IV. New phases and boundaries indicated by dashed black lines are still speculative: the last portion of the melting line and the IV-IV’ line is proposed in [5]; phase V and the associated transition line is proposed by [17]; phase VI is observed by [18] and [19], even

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1.1.1 Phase I

At relatively low pressures and temperatures, interactions are weak and the molecular angular momentum J can be regarded as a good quantum number: we can then talk about para-hydrogen p − H2 (even J ) or ortho-hydrogen o − H2 (odd J ) as we usually do when dealing with isolated

molecules. Since transitions between odd and even J states are prohibited for isolated molecules, conversion from one species to another is very slow at low pressures and the two species can be studied separately or in mixtures with constant concentrations.

At room temperature and atmospheric pressure, hydrogen is a diatomic gas; at first, solid hydrogen was obtained at approximately zero pressure by cooling the system at helium temperatures. Early X-ray diffraction studies showed that at these temperatures the molecular centers were arranged according a hexagonal close packed structure (hcp) regardless of the ortho-para ratio [20–22]. Using diamond anvil cells, solid hydrogen was observed at room temperature [9] at 5.7 GPa; subsequent X-ray diffraction studies were performed, reaching pressures of tens of GPas [23,24], finding the same hcp structure reported at zero pressure and helium temperatures. These results can be integrated with observations coming from Raman and infrared spectra experiments, which are successfully used to probe rotational and vibrational properties of molecular systems. For high pressure hydrogen, these spectra can be consistently interpreted assuming the anisotropic part of the intermolecular interaction to be negligible. In this case, in fact, only the molecular radial coordinates are coupled, resulting in a collection of independent quantum rotors [22, 25]. Para-hydrogen molecules, being in a J = 0 ground state, have spherical symmetry; on the other hand, the J = 1 degenerate states of ortho-hydrogen molecules are equally populated, producing a spherical symmetry also in this case. The consistency of this approach is verified at low pressures and temperatures, obtaining sharp rotovibrational Raman peaks [26, 27]: this indicates that the mixing of different rotational levels is small. At higher temperatures and pressures, a continuous broadening of the rotational peaks takes place [28], corresponding to an increasing strength of the neglected anisotropic interaction, which could drive the system towards a rotationally ordered system. This solid phase of free quantum rotors is known as phase I.

1.1.2 The broken symmetry phases: phase II and III

Measurements of Raman and infrared spectra can be performed beyond the pressure and temperature range imposed by X-ray diffraction, probing vibrational properties of the system in a larger portion of the phase diagram. Vibrational properties are sensitive to changes in crystal structure or rotational order: the region of stability of phase I can thus be naturally prolonged as long as the vibrational spectra do not display significant changes; on the other hand, discontinuities in these quantities are related to phase transitions.

This is the case, for example, for deuterium and hydrogen below 140 K in a pressure range of 40-150 GPa: the position of the Raman and IR peaks associated with the vibration of the molecular bond (typically referred to as vibrons) shows a discontinuity of tens of cm−1; at the same time, a cusp is observed in its temperature dependence and several changes take place in the low-frequency spectrum [4, 29–35]. In particular, the appearance of new peaks at low frequency is associated to the presence of librons, phonons associated to a restricted orientational motion: for this reason

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Figure 1.2: Phase boundaries of low temperature solid phases for normal hydrogen and deuterium, based on optical Raman and IR spectra. Picture from [29].

the transition to the new phase, called phase II, is supposed to involve orientational ordering, even for spherical J = 0 species (para-hydrogen and ortho-deuterium). This is consistent with the expectation that at high enough pressures the electric quadrupole-quadrupole interaction would mix different rotational states, resulting in an ordered, anisotropic state.

The boundaries of phase II, based on optical spectra, are depicted in fig. 1.2 for normal H2 and

D2 with an equilibrium ortho-para concentration. While Raman and IR experiments signal the

transition, they can at best provide constraints for the symmetry of the structure. An early candidate for phase II was the P a3 structure, where the molecular centers are on the sites of a

face centered cubic crystal while the molecules are oriented according to a specific pattern; this is the structure found for the orientational ordered phase of ortho-hydrogen and para-deuterium at ambient pressure and T < 3 − 4 K [30]. Unfortunately, a group theory analysis of the number of Raman and IR peaks deemed this structure incompatible with experimental results [36]. Recent X-ray diffraction and neutron scattering studies [15, 16] suggest that the hcp lattice is retained, with molecules being locally oriented similarly to P a3.

Following the argument of increasing anisotropic interaction when increasing pressure, a break of the rotational symmetry is expected even at higher temperature. This is indeed the case: a transition to a new phase, phase III, is detected for hydrogen around 150 GPa. The transition, however, takes place at high and low temperatures, starting from both phase I and II (see fig. 1.2): it seems to involve more than only rotational ordering. Similarly to the I-II transition, the I-III and II-III transitions are characterized by a discontinuity in the vibron frequency (that can be of hundreds of cm−1), a change in the slope of the vibron dependence on temperature and

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a change of the low frequency spectrum, even larger than in the I-II transition [29, 30, 34]. The more pronounced features may correspond to a larger structural change. Moreover, a peculiarity of transitions to phase III is the sudden increase in IR absorption (three orders of magnitude) for the vibronic peak [4, 18, 33], which hints at a primitive cell with a large (compared to the other phases, at least) dipole moment. An X-ray diffraction experiment [15] found that molecular centers in phase III remain close to the hcp lattice sites, at least up to 183 GPa; no direct information about molecular orientation could be extracted. A classical orientational ordering process is speculated to take place [37]. Phase III is transparent [4, 38] and insulating in the experimentally accessible thermodynamic conditions. Phases I, II, and III coexist at a triple point at 125 K and 155 GPa for H2 (135 K and 165 GPa for D2)

1.1.3 Mixed phases: phase IV and IV’

In 2011, Eremets et al. [39] reported the observation of a semiconducting phase of hydrogen at T=295 K above 220 GPa, using photoconductivity and resistance measurements; a metallic state was claimed to be produced above 260 GPa. At the same time, significant changes in the low frequency Raman peaks take place at the transitions. Subsequent Raman and IR studies [5, 40–42] show that when the first transition occurs at 220 GPa, a new high-frequency peak appears at ≈ 4150 cm−1, next to the vibron peak present in phase III, which displays a discontinuity of hundreds of cm−1 _{and softens very rapidly after the transition. The presence of two high frequency peaks is}

interpreted as the existence of two different local atomic environments: in particular, theoretical calculations [43] suggest the possible stability of layered mixed structures, where atoms belonging to different layers are bonded in a different manner. This topic will be treated in more detail in chapter 3. More recent optical measurements [38, 41] contradict the reported metallicity; semimetallic or semiconducting behaviour is still not ruled out. This new phase is labelled as phase IV. At the same time, the change in intensity and number of the low frequency phononic peaks at higher pressure was confirmed [5, 41] and the new phase resulting from the transition is labelled as IV’ or V. The experimental boundaries of the distinct five solid phases are pictured in fig. 1.1.

1.1.4 Possible phases at higher pressures

With the improvement of DAC techniques, the experimental exploration of the solid region of hydrogen phase diagram is still an on-going activity, reaching higher and higher pressures. Different experiments display conflicting results. Eremets et al. [18] study hydrogen at pressures up to 380 GPa and T<200 K with Raman scattering. For P>360 GPa they find that the intensity of the Raman spectra goes to zero when cooling the system below 200 K; at the same time, a strong drop in resistance is observed in the same thermodynamic conditions (P>360 GPa and T<200 K). They thus draw a vertical transition line in the P-T plane, introducing a new conducting phase VI for pressures higher than P=360 GPa. Dalladay-Simpson et al. [17] investigate the system at T ≥ 300 K. They propose a new phase (V) for P>325 GPa, based on arguments similar for phase transitions at lower pressures: change in the low frequency peaks, change in the slope of the pressure dependence of the vibron, broadening and weakening of the vibrational peak itself. The Raman intensity, in general, decreases: this, coupled to the weak vibronic signal, is interpreted as

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quasi-atomic state, precursor of a fully non molecular, metallic system. Dias et al. [19] probe the system at low temperature (T<200 K), as in [18], using infrared radiation. Above 355 GPa, the IR vibron disappears, and two new peaks close to 3000 cm−1 appear; they come up with a vertical transition line, similar to [18]. At variance with [18], though, no evidence of metallicity is found. As we can see, the region of the phase diagram at low temperature (T ≈ 200 K) and high pressure (P>300 GPa) displays interesting phenomena, either the long-yearned-for metallic transition or a “simple” structural transition; moreover, we must not forget that even the crystal lattice corresponding to phase III, which is relatively well established in the solid phase diagram, is still an object of debate. At higher temperatures, experiments point to the stability of mixed structures, whose finite temperature properties may be difficult to predict. These will be the starting points of our discussion in chapter 3.

1.2 Liquid-liquid phase transition

As anticipated in the first paragraph, metallic hydrogen was first produced in the liquid state [6]: using multiple shock compression, liquid hydrogen was compressed to 140 GPa and heated to 3000 K; a sharp drop in resistivity (≈ 500µΩcm) was observed, a clear signal of the metallic state. Nevertheless, a clear characterization of the insulator-metal (IM) transition is still missing and many questions remain open. Traditionally, dynamic compression experiments are plagued by large uncertainties: for example, temperature is usually estimated through equations of state and not actually measured, introducing systematic source of errors. This is apparent in fig. 1.3 for the yellow and maroon signs. The improvement in static compression using DAC allows to perform such experiments in the liquid phase as well (green and purple signs for hydrogen in fig. 1.3). In particular, laser heated experiments are performed: the sample is usually in contact with a metallic absorber embedded in the diamond anvil cell; the absorber is heated through laser pulses, and the sample is heated by direct contact. The natural effect is that the temperature of the sample rises; Dzyabura et al. [47] found that the temperature does not increase indefinitely while heat is provided, but a plateau is reached. This can be interpreted as latent heat, provided to make a first-order phase transition happen. Rigorous evidence of metallization, however, required a systematic study of optical properties, which could not be performed due to mechanical instabilities. Ohta et al. [11] performed similar experiments at higher temperatures (T>2000 K, see fig. 1.3), finding the same saturation of the maximum reachable temperature. In particular, they assume that the transition under examination is the molecular dissociation of liquid hydrogen; while stating that this could be related to the insulator-metallic transition, they do not provide any measurement of optical quantities.

Zaghoo et al. [10] repeated the laser-heating experiments, integrating the temperature plateaus with measurements of optical reflectivity. Reflectivity, which may provide indications about the metallic state of the system, was measured using a second laser pulse to probe the system. In particular, they start from a transparent sample and, while heating, they measure an increase in reflectivity, that saturates to 0.5. As we can see from fig. 1.3, their results for the liquid-liquid phase transition are compatible with Ohta’s; measuring the reflectivity, they provide evidence to link molecular dissociation with metallization of hydrogen.

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100 200 300 Pressure (GPa) 0 1000 2000 3000 Temperature (K)

Fluid H

₂

Fluid H

Solid H

₂ III I II IV CEIMC Knudson 2015 Weir 1996 Zaghoo 2015 Fortov 2007 IV’ Ohta 2015

Figure 1.3: Hydrogen phase diagram with experimental liquid–liquid transition lines and CEIMC predictions. Blue circles and squares are CEIMC predictions for the liquid–liquid transition line in hydrogen and in deuterium, respectively. DAC experimental results for hydrogen are indicated by green circles [10] and purple circles [11], whereas red squares indicate shock wave experimental data for deuterium [14]. We also report an earlier experimental point for the conductivity onset in liquid hydrogen (yellow circle) [6] and two points for deuterium dissociation and metallization (maroon squares) [44]. The error bars on the temperature for the latter two sets of data reported here were inferred from theories [45]. In fact, all dynamical experiments [6, 14, 44] used models to determine T. The blue triangle at 3000 K indicates the CEIMC prediction for a metallization cross-over above the critical point where the conductivity is 2 − 4 × 103_(Ωcm−1

). Figure from [46]

Reflectivity measurements for deuterium by Knudson et al. [14] using dynamical compression find a transition line which is 150 GPa apart from Zaghoo’s findings; an isotopic effect is not probable, since previous dynamical compression experiments on deuterium by Fortov [44] found a transition point closer to Zaghoo’s line.

A challenging problem with optical measurements is that the different parts of the experimental apparatus must be considered when extracting the sample response from the raw data. Moreover, hydrogen is assumed to have perfect transmittance and zero reflectivity at the wavelength of the probe before the transition; an hypothesis that will be discussed in chapter 4, where we will show our results for the liquid-liquid phase transition, obtained from Ab Initio simulations. We will also provide optical properties, that may help in disentangling the actual contribution of the sample from the rest of the apparatus.

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Simulation methods

Ab Initio computer simulations are a valuable tool to study condensed matter systems. They can cooperate in synergy with experiments, helping in the interpretation of experimental data and providing missing information. But they can also achieve predictive power when exploring new territories, working as an input to design new experiments. Currently, the most popular Ab Initio method is Density Functional Theory (DFT). While the theoretical foundations of the method were laid in 1964 by Hohenberg and Kohn [48], its practical implementation proposed in 1965 by Kohn and Sham [49], it was in the 1990s that DFT became extremely popular : DFT computations on small molecular systems employing hybrid exchange-correlation functionals (such as B3LYP) outperformed other more computationally expensive Ab Initio methods (Hartree-Fock, Self Consistent Field) when compared against experiments [50, 51]. However, even DFT has limitations which will be described in the following sections. In this chapter we give an overview the computational challenges inherent to high pressure hydrogen and introduce the techniques used to deal with them.

2.1 The electronic problem and the Born-Oppenheimer

ap-proximation

We consider a system of Ne, spin unpolarized, electrons and Np protons in a fixed volume Ω in

thermal equilibrium at temperature T . In atomic units, the Hamiltonian of the system is

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where ~ri and ~Rα are positions for the i-th electron and the α-th proton respectively. This leads to

the following time-independent Schrodinger equation written in the electronic and nuclear position basis set:

ˆ

H(R, r)Φk(R, r) = EkΦk(R, r) (2.4)

where R and r stand for the collection of all nuclear and electronic coordinates, respectively. Solving eq. 2.4 even for small molecules is beyond current computational capabilities: at this point the Born-Oppenheimer approximation [52–54] is introduced to make the problem tractable. We define the wavefunctions Ψn(r; R) as

ˆ

HeΨn(r; R) = ( ˆKe+ ˆV)Ψn(r; R) = Ene(R)Ψn(r; R) (2.5)

i.e. they are eigenfunctions of the operator ˆHe. Since ˆHe contains the nuclear coordinates in the

potential term, both the eigenvalue Ene(R) and the eigenfunction Ψn parametrically depend of

the nuclear coordinates. Expanding the total wavefunction Φk(R, r) as a linear combination of

wavefunctions Ψn(r; R)

Φk(R, r) =

X

n

χkn(R)Ψn(r; R) (2.6)

and plugging eq. 2.6 in eq. 2.4 we obtain

X n ˆ Kp(χkn(R)Ψn(r; R)) + X n χkn(R)Ene(R) Ψn(r; R) = X n Ekχkn(R)Ψn(r; R) (2.7)

The action of the nuclear kinetic operator on the total wavefunction produces the following terms:

ˆ Kp(χkn(R)Ψn(r; R)) = − 1 2Mp Nα X α ∇2 Rα(χkn(R)Ψn(r; R)) = = − 1 2Mp Nα X α ∇2 Rαχkn(R) Ψn(r; R) − 1 2Mp Nα X α ∇2 RαΨn(r; R) χkn(R) − (2.8) − 1 Mp Nα X α (∇Rαχkn(R)) · (∇RαΨn(r; R))

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Multiplying eq. 2.7 by Ψm(r; R), integrating the electronic coordinates and rearranging the terms, we get ( ˆKp+ Eem(R) − Ek)χkm(R) = 1 2Mp X n Nα X α χkn(R) Z drΨm(r; R)∇2RαΨn(r; R) + + 1 Mp X n Nα X α ∇Rαχkn(R) · Z drΨm(r; R)∇RαΨn(r; R) (2.9)

If we ignore the r.h.s. of eq. 2.9, we obtain an eigenvalue problem for the nuclear coordinates corresponding to a “clamped nuclei” Hamiltonian ˆHcn= ˆKp+ Eme(R) with a potential given by the

electronic eigenvalue Ee

m(R): this is the Born-Oppenheimer approximation. The terms on the r.h.s.

provide both adiabatic and non adiabatic corrections, coupling different electronic eigenstates. If there are no magnetic fields, we can choose to work with real wavefunctions and it is easy to prove by integration by parts that diagonal terms likeR drΨm(r; R)∇RαΨm(r; R) are identically zero. It

can be shown that the other non diagonal terms are [54]

hΨm|KpΨni ∝ 1 Em− En hΨm| h −i∇Rα, ˆH i Ψni (2.10)

The numerator depends only mildly on the nuclear coordinates: thus, if the electronic energy surfaces are well separated, the non diagonal terms are negligible because of the denominator. However, this is not valid for metals, which are an important class of systems: one of the motors driving research on high pressure hydrogen is predicting when the system becomes metallic. In this case, an argument can be made that the most relevant excitations are single-electron ones, which are barely coupled to the nuclear motion [54] . Collective excitations, such as plasmons, are of high energy (some eVs) and the previous line of reasoning stays true.

In this approximation, the original problem can be simplified and resolved in the following way:

• (i) given the configuration R, solve eq. 2.5 and find the electronic ground wave function Ψ0(r; R)

• (ii) assuming that the electrons are in the ground state, use Ψ0(r; R) to solve

ˆ_K_p_{+ E}e

0(R) − E0

χk0(R) = 0

and find the nuclear wavefunction χk0(R)

The above procedure can be followed to obtain electronic and nuclear wavefunctions for pure states. When different quantum states have finite probabilities of being occupied, density matrices are employed [55]:

ˆ ρ =X

s

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where ps is the probability of finding the system in the state |Φsi. If we work in the canonical

ensemble, the density matrix is ˆ ρc = X s e−βEs_|E si hEs| = e−β ˆH (2.12)

where |Esi is the an eigenstate of the total Hamiltonian. The physical properties of the system are

obtained as statistical averages

O = h ˆOi =

Trh ˆO ˆρc

i

Tr [ ˆρc]

(2.13)

over the measure given by the thermal density matrix ˆρc. In the spirit of the Born-Oppenheimer

approximation we assume that electrons are in their ground state, while nuclei are at finite temperature. This corresponds to writing the partition function Z of the system as

Z = Tr [ˆρc] ≈

Z

dR hR| hΨ0| e−βH|Ψ0i |Ri (2.14)

The matrix element hΨ0| e−βH|Ψ0i can be written in a Taylor expansion:

hΨ0| e−βH|Ψ0i = X l (−β)l l! hΨ0| ˆ_K_p_{+ ˆ}_H_el |Ψ0i (2.15)

The single term can be evaluated:

hΨ0| ˆKp+ ˆHe l |Ψ0i = hΨ0| ˆKp+ ˆHe l−1 ˆ_K_p_{+ ˆ}_H_e |Ψ0i = hΨ0| ˆKp+ ˆHe l−1 P l0|Ψl0i hΨ_l0| ˆK_p+ ˆH_e |Ψ0i ≈ ≈ hΨ0| ˆKp+ ˆHe l−1 P l0|Ψl0i hΨ_l0|Ψ₀i ˆK_p+ Ee₀(R) = hΨ0| ˆKp+ ˆHe l−1 |Ψ0i ˆKp+ E0e(R) (2.16)

We ignored the action of the nuclear kinetic operator ˆKp on the electronic wavefunction Ψ0, which

is consistent with neglecting the terms in the r.h.s. of eq. 2.9. Iterating the process, one can easily prove that the density matrix in the Hilbert space of the nuclear degrees of freedom reduces to

hΨ0| e−βH|Ψ0i ≈ ˆρBO∝ e−β( ˆKp+E

e

0(R)) _(2.17)

In this way the trace can be written as the sum over different nuclear configurations only, with the potential energy surface Ee

0(R) which plays the role of the effective interaction among nuclei.

Any ab-initio method based on the BO approximation needs to address the problem of calculating E₀e(R) while keeping the nuclear coordinates fixed. Resolving the many-body electronic Schrodinger equation for large extended systems still remains a computational challenge with the resources currently available. As already remarked, nowadays the most widespread Ab Initio method is Density Functional Theory (DFT), whose theoretical foundations were laid in 1964 by Hohenberg and Kohn [56].

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2.2 Density Functional Theory

To solve eq. 2.5 for a system with Neelectrons, one must deal with the 3Neelectronic coordinates.

Any finite element method based on a numerical discretization of the many-body wavefunction is doomed to fail for extended systems with tens, hundreds of particles due to the exponential use of memory necessary to store the variables. Density functional theory provides an alternative path: instead of considering the electronic ground state wavefunction Ψ0(r; R), the main quantity in this

approach is the ground state single electron density

n0(~r) = Ne

X

i=1

hΨ0|δ(~ri− ~r)|Ψ0i (2.18)

From a theoretical point of view, knowledge of n0(~r) proves to be sufficient to derive any other

ground state property of the system as a functional of n0(~r) itself (in particular, the ground state

energy Ee

0= Ee[n0]). Computationally, this resolve the issue of dealing 3Nevariables, reducing it

to a three dimensional problem. On the other hand, an explicit, exact functional form is missing, forcing any actual implementation of DFT to make use of uncontrolled approximations.

In the next section, a proof of the Hohenberg-Kohn theorem, which establishes the formal connection between n0(~r) and any other ground state quantity, will be given.

2.2.1 The Hohenberg-Kohn theorem

To give a more general scope to the theorem, the electronic Hamiltonian is rewritten as ˆ He= ˆKe+ ˆV + ˆW (2.19) where ˆ Ke= − 1 2 Ne X i=1 ~ ∇2 ~ri , ˆ V = Ne X i=1 v(~ri) , W =ˆ Ne X i=1,j>i 1 |~ri− ~rj| (2.20) ˆ

Ke is the usual kinetic energy. ˆV is the external potential, which can be written as a sum of

one-particle terms. In particular, its average value over the ground state wavefunction |Ψ0i can be

In the Born-Oppenheimer approximation, this is generally the potential generated by the nuclei but it can also include any other external field. Finally, ˆW is the coulombic repulsion among different electrons. Note that, within the class of Hamiltonians defined in eq. 2.19, the only difference is

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the external potential ˆV: thus, any observable A can be seen as A[v], a functional of v(~r), since everything else is fixed.

The theorem can be enunciated as follows [48, 57, 58]:

HK theorem:. (i) The ground state electronic density n(~r) uniquely determines the external potential v(r), to within an additive constant, i.e. v(r) is a unique functional of n(~r) (ii) The ground state energy E0 is the minimum of the functional E[n] for all v-representable

densities n(~r)

Not any electronic density n(~r) one can come up with can be thought as the ground state density of some Hamiltonian that has the form defined in eq. 2.19. A “v-representable density n(~r)” enforces precisely this restriction.

The first statement can be proved in the following way, using reductio ad absurdum: we suppose that two potentials ˆV and ˆV0 _{corresponding to the same n}

0(~r) exist and ˆV 6= ˆV0+ const. We then

have two Hamiltonians ˆH = ˆKe+ ˆV + ˆW and ˆH0 = ˆKe+ ˆV0+ ˆW and two ground state wavefunctions

|Ψ0i and |Ψ00i with energies E0e and Ee

0

0 , First of all, we show that the two Hamiltonians cannot

have a common ground state, i.e. |Ψ0i 6= |Ψ00i. If that were the case and |Ψ0i = |Ψ00i,

( ˆH − ˆH0) |Ψ0i = ( ˆV − ˆV0) |Ψ0i = Ne X i=1 (v(~ri) − v0(~ri))Ψ0(r) = (E0e− E e0 0 )Ψ0(r) (2.22)

i.e. v(~r) − v0_(~_{r) = const when the wavefunction does not vanish. It can be shown [58] that}

for reasonable forms of v(~r) this implies v(~r) − v0(~r) = const everywhere, which contradicts the hypothesis: the two ground states |Ψ0i and |Ψ00i must be distinct. Notice that the ground states of

and the two inequalities holds strictly since we know that |Ψ0i and |Ψ00i cannot simultaneously be

ground state for both Hamiltonians. Now, subtracting the two inequalities in eq. 2.23 and applying eq. 2.21 , we obtain E₀e− Ee0 0 < E e0 0 − E e 0+ hΨ00| ˆV − ˆV0|Ψ00i − hΨ0| ˆV0− ˆV|Ψ0i = = E₀e0− E0e+ Z d~r (v(~r) − v0(~r)) n(~r) − Z d~r (v0(~r) − v(~r)) n(~r) = = E₀e0− Ee 0 (2.24)

where we used the assumption that the two potentials correspond to the same ground state density n(~r), which leads to a contradiction.

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The main accomplishment of the first statement of the theorem is that every functional dependence on v is shifted on n, thanks to the unique functional correspondence v[n]. This completely determines the Hamiltonian, that isH[v] = ˆˆ H[v[n]]. Similarly, once the Hamiltonian is fixed, the quantum mechanical problem can in principle be solved and every property traced back to n (even excited states). In particular, |Ψ0i is a “natural” functional of the potential v and, by composition, must

also be a functional of the ground state density. The ground state energy can be written as

E₀e[v[n]] = hΨ0[v[n]]| ˆH[v[n]]|Ψ0[v[n]]i = F [n] + Z d~rv[n](~r)n(~r) (2.25) F [n] = E₀e[v[n]] − Z d~rv[n](~r)n(~r)

where we isolated F [n], the Kohn-Sham functional. Some remarks:

• while v[n] is a unique functional, n[v] and |Ψ0[v[n]]i are unique only if the ground state is

non degenerate • Ee

0[v[n]] is unique, being the composition of two unique functionals.

• F [n] is unique and well defined as well (the arbitrary constant in v(~r) cancels out) • in the non degenerate case, the functional F [n] can be defined as

hΨ0[n]| ˆT + ˆW|Ψ0[n]i (2.26)

since all the mappings are unique. This corresponds to the definition given in eq. 2.25 which is, however, more general.

We can show that the variational principle holds with respect to the functional dependence on the ground state density. To prove it, the following functional, which linearly depends on the external potential, is introduced:

Ev[n] = F [n] +

Z

d~rv(~r)n(~r) (2.27)

where we treat v and n as two independent variables (i.e. the potential v is not necessarily vn= v[n]).

Manipulating this functional, we obtain

The true ground wavefunction for the Hamiltonian ˆH[v] is Ψ0[v] ≡ Ψ0[v[n0]], where n0is the ground

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principle states that

Ev[n] = hΨ0[vn]| ˆH[v]|Ψ0[vn]i ≥ hΨ0[v[n0]]| ˆH[v]|Ψ0[v[n0]]i = Ev[n0] (2.29)

i.e. the variational principle holds using the ground state density as the main independent variable; the equality is obtained if we plug into the expression exactly n0.

2.2.2 The Kohn-Sham equations

While the HK theorem establishes the ground state electronic density as the main variable, it does not provide an explicit form for E[n]; in particular, the universal functional F [n], which is the non-trivial part of E[n], is unknown. Moreover, F [n] is the same for any electronic system: a direct guess for such a complicated object is very unlikely. To make use of Density Functional Theory, practical implementations for these elements are needed: in the following paragraphs we will obtain equivalent expressions for the two functionals, which can (mostly) be evaluated in a simple manner. We start [49, 57, 58] considering a non interacting system (i.e. W = 0) with Hamiltonian

ˆ

H = ˆT + ˆVs (2.30)

where ˆVs = Pivs(~ri) is an external potential. If we assume non-degeneracy, the ground state

wave-function Ψ0_s({~ri}) is a Slater determinant made of orthonormal single particle orbitals ψk(~ri)

Ψ0_s(r) = √1

N !det [ψk(~ri)] (2.31)

and the individual orbitals can be obtained by resolving the following single-particle equations: −1 2∇ 2 + vs(~r) ψk(~r) = εkψk(~r) (2.32)

and then pick the Ne orbitals with the lowest energies εk to form Ψ0s(r) The ground state density

and the kinetic energy are respectively

ns(~r) = occ X k ψ_k∗(~r)ψk(~r) Ts= − 1 2 X k occ Z d~rψ∗_k(~r)∇2ψk(~r) (2.33)

The demonstration in the previous section does not require any specific W (as long the Hamiltonian is bounded from below). If we apply our results to the non interacting system, we obtain

Es[n] = Ts[n] +

Z

d~rvs[n](~r)n(~r) (2.34)

When there is no interaction Fs[n] = Ts[n]. The KS variational principle holds, and Evs[n] will be

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The non interacting system just introduced will act as an effective system for the interacting problem we are effectively interested in. For that, we know that E[n], given in eq. 2.27 will be a minimum when evaluated at n0(~r), the interacting ground state density. We now assume that, given n0(~r),

we can build a non interacting system with an appropriate vs(~r) such that ns(~r) = n0(~r). To do

this, we write E[n] as

E[n] = Ts[n] + EH[n] + Eext[n] + Exc[n] (2.35) where EH[n] = 1 2 Z d~rd~r0n(~r)n(~r 0₎ |~r − ~r0_| (2.36) Eext[n] = Z d~rvext(~r)n(~r) (2.37) Exc[n] = F [n] − Ts[n] − EH[n] (2.38)

EH[n] is the Hartree term, accounting for the electron-electron interaction; Eext[n] is the energy

functional coming from the external potential; finally, the definition of Exc[n] is a tautology. The

first two functionals can be easily evaluated once the density is known. On the other hand, at this stage, we did not gain anything: the explicit form of Exc[n], which includes all the complicated

many-body effects, is as unknown as the functional F [n]. The hope is that this term should be small compared to the others for most systems, and that approximations could work reasonably well in this regime.

We now use the variational principle, imposing:

E[n0+ δn] − E[n0] = O(δn2) (2.39)

Evaluating the single terms we have:

Eext[n0+ δn] − Eext[n0] = Z d~rvext(~r)δn(~r) + O(δn2) (2.40) EH[n0+ δn] − EH[n0] = Z Z d~rd~r0δn(~r) 1 |~r − ~r0_|n0(~r 0_{) + O(δn}2₎ _(2.41) Exc[n0+ δn] − Exc[n0] = Z d~rδExc[n] δn(~r) |n=n0δn(~r) + O(δn 2₎ _(2.42) Ts[n0+ δn] − Ts[n0] = − 1 2 X k occ Z d~rδψ_k∗(~r)∇2ψk(~r) + δψk(~r)∇2ψ∗k(~r) + O(δψ 2_)(2.43)

The last equation can be evaluated using eq. 2.32 and writing explicitly the variation δn: δn(~r) = X k occ δψ_k∗(~r)ψk(~r) + ψk∗(~r)δψk(~r) + O(δψ2) (2.44) Ts[n0+ δn] − Ts[n0] = X k occ Z d~r [εk− vs(~r)] (δψ∗k(~r)ψk(~r) + δψk(~r)ψ∗k(~r)) + O(δψ 2_{) =} = − Z d~rvs(~r)δn(~r) + O(δn2) (2.45)

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In the last step, we used the orthonormalization of the wavefunction to get rid of the εk term: in fact Z d~r (δψ_k∗(~r)ψk(~r) + δψk(~r)ψk∗(~r)) = δ Z d~r|ψk(~r)|2= 0 (2.46)

Putting all together and imposing eq. 2.39, we obtain Z d~r vs(~r) − Z d~r0 1 |~r − ~r0_|n0(~r 0_{) − v} ext(~r) − δExc[n] δn0(~r) |n=n0 δn(~r) = 0 (2.47) i.e. vs(~r) = vH[n0](~r) + vext(~r) + vxc[n0](~r) (2.48) vH[n0](~r) = Z d~r0 1 |~r − ~r0_|n0(~r 0₎ _(2.49) vxc[n0](~r) = δExc[n] δn0(~r) |n=n0 (2.50) (−1 2∇ 2₊ _v H[n0](~r) +vext(~r) + vxc[n0](~r))ψk(~r) = εkψk(~r) (2.51)

These are the Kohn-Sham equations [49]: they are non linear, since vH[n0] and vxc[n0] depend on

the ground state density (and thus on the orbitals ψk(~r)) and are usually solved by an iterative

procedure starting from an initial guess for the orbitals. At this point we stress that as Exc[n]

is unknown, so is vxc[n0]. Approximations can be built from physical intuition, but there is no

systematic way to build and improve them: they are uncontrolled. There are different forms of Exc[n] that can be more suitable for a particular system, but most of the times this cannot be

determined a priori: comparison with experiments or more rigorous theories (when available) is necessary.

2.2.3 Practical implementation

Due to computational limitations, relatively small cells (≈ 100 atoms in our case) are employed to simulate extended systems. Given a cell defined by three vectors (~L1, ~L2, ~L3), it is common

practice to use periodic boundary conditions, periodically repeating the simulation cell. When dealing with crystal structures, the simulation cell is a supercell, containing several primitive cells. The translational symmetry of the system ensures that Bloch’s theorem [59] holds, i.e. that KS orbitals can be written as

ψ~k(~r) = e i~k·~r_u

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being decomposed in a periodic function modulated by the wave ei~k·~r_{. Introducing the reciprocal}

lattice vectors ~Gm~, defined as

~ Gm~ = m1B~1+ m2B~2+ m3B~3 , m = (m~ 1, m2, m3) ∈ N3 (2.53) ~ B1 = 2π ~ L2× ~L3 ~ L1· (~L2× ~L3) ~ B2 = 2π ~ L3× ~L1 ~ L2· (~L3× ~L1) (2.54) ~ B3 = 2π ~ L1× ~L2 ~ L3· (~L1× ~L2)

the orbitals can be represented as ψ~k(~r) = e i~k·~r_u ~k(~r) = e i~k·~rX ~ m C~k ~me i ~Gm·~r _(2.55)

where the summation over integers is possible thanks to the periodicity of uk(~r). Moreover, it is easy

to verify that for any vector ~Gm~ ψ~_{k+ ~}_G

~

m(~r) = ψ~k(~r): the vector ~k can be confined to the primitive

cell of the reciprocal lattice, conventionally the first Brillouin zone. This plane-wave expansion is used in many codes written to deal with periodic systems [60–62] and it is particularly convenient since Fourier transforms can be performed using efficient algorithms that scale as Npwlog(Npw)

where Npw is the number of coefficients Ckmconsidered. When recast in the plane wave basis set,

eq. 2.51 is X ~ m0 1 2 ~ k + ~Gm~ 2 δm, ~~ m0+ ˜v_H( ~G_m_~ − ~G_m_~0) + ˜v_ext( ~G_m_~ − ~G_m_~0) + ˜v_xc( ~G_m_~ − ~G_m_~0) C_kmi 0 = εi_kC_kmi(2.56) with ˜ vl( ~G) = _Ω1 R_Ωd~rvl(~r)ei~k· ~G , l = H, ext, xc (2.57)

In fact, for every ~k we have a different Schroedinger equation: each equation has its own solutions, labelled by the i index, corresponding to a wavefunction ψi

~_k(~r).

Since computer memory is finite, an inevitable approximation is truncating the sum in 2.55: a cutoff is usually defined as

1 2 ~_G ~ m+ ~k 2 ≤ Ecut (2.58)

The value of Ecut depends on the behaviour of the wavefunction in the proximity of one of the

nuclei. In the sections regarding Quantum Monte Carlo it will be shown that the divergence of the Coulomb potential near the origin introduces a “cusp” in the wavefunction:

∂ψk

∂r |r=0= −Zψ

00

k (2.59)

where ψ00_{represents the spherical average of the wavefunction around the nucleus and r is the radial}

coordinate relative to its position. The cusp needs many terms in the planewave expansion to be accurately represented and, thus, a high value of Ecut. To circumvent the problem, pseudopotentials

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are introduced [63]: a cut-off radius rc is introduced and, within the sphere delimited by rc, the full

electron-nucleus Coulomb interaction is replaced by a smooth potential, generated to reproduce relevant properties of the “true” isolated atom. For heavier elements, another advantage of using pseudopotentials consists in eliminating core electrons, that are not relevant for chemical binding. Different strategies can be used: in our computations we either used a Coulomb potential when producing single particle orbitals for the Monte Carlo trial wavefunction (see following sections for details) or a PAW pseudopotential [64], which gives the most accurate results with a low value of Ecut.

With the introduction of the vector ~k, many quantities (such as densities, energies) can be expressed as hOi = Ω (2π)3 X i occ Z BZ d~kO_~i_k (2.60) O_~i k = hψ i ~_k|O|ψ i ~ ki (2.61) where R

BZ stands for an integral over the first Brillouin zone. Inevitably, integrals like this one

must be reduced to sums to be evaluated computationally:

hOi = 1 Nk X k∈BZ, i occ O_~i k (2.62)

with Nk being the number of ~k points sampled. A typical choice is to take a grid of vectors ~k

defined as ~_{k =} n1+1₂ N1 ~ B1+ n2+1₂ N2 ~ B2+ n3+1₂ N3 ~ B3 (2.63)

where ni,Ni are integers and −Ni/2 ≤ ni< Ni/2, for a total of N1× N2× N3points. This is the

so called Monkhorst-Pack mesh [65].

Finally, the crucial approximation: the choice of Exc. In our calculations two different Excwere

employed: PBE [66], that is based on a parametrization of the homogeneous electron gas, and vdW-DF [67], that focuses on capturing the physics of systems where the van der Waals interaction is relevant. These choices will be justified in the appropriate sections.

2.3 Quantum Monte Carlo

As explained in the previous section, the practical implementation of DFT is plagued by the problem of approximating Exc[n]. Nevertheless, it reduces the complexity of the many body

quantum problem bypassing the evaluation of the electronic wavefunction. As mentioned at the beginning of this chapter, a discretization on a grid of the many body wavefunction is unfeasible; nevertheless, we will show how the wavefunction can be associated to a probability density that can be sampled using stochastic Monte Carlo methods. In this way there is no need to actually store the huge amount of information contained in the wavefunction and any physical observable can be computed as an average over that probability density. We will start describing the basics of Monte Carlo methods.

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2.3.1 Monte Carlo methods and the Metropolis algorithm

In many applications, an important problem is how to efficiently compute quantities that can be written as

hOi = Z

dxO(x)p(x) (2.64)

where p(x) is a correctly normalized probability density (i.e. p(x) ≥ 0 ∀x andR dxp(x) = 1) and x is a multidimensional array of continuous or discrete variables. When the number of dimensions of the x-space grows, a discretization of the integral based on finite differences requires a mesh of exponential size; on the other hand, if we can sample configurations xi according to p(x), a good

estimate of hOi is hOi ≈ ¯O = 1 Nc Nc X i=1 O(xi) (2.65)

A key feature of this approach is that the error associated to ¯O is

σO¯ = s σ_O,p2 Nc (2.66) σ2_O,p = 1 Nc− 1 Nc X i=1 O(xi) − ¯O 2 (2.67) σ2

O,pdepends on the observable O and on the probability distribution p(x) and it is a fixed feature of

the process we are studying. This means that the error σ_O¯ ∝√1

Nc irrespective of the dimensionality

of x; this is a huge advantage when dealing with high dimensional arrays. To compute ¯O one must be able to efficiently sample xi according to p(x): this is accomplished using Markov chains [68].

A Markov chain is a stochastic process where configurations are generated in a sequence, and the probability of having a state xiat step ti depends only on the configuration at step ti−1

PC(xi, ti|xi−1, ti−1; xi−2, ti−2; . . . , x1, t1) = PC(xi, ti|xi−1, ti−1) (2.68)

where PC(xi, ti|xj, tj) is the conditional probability of having xi at step ti given xj at step tj.

Given a target probability density Ptarget(xi), the purpose of the Metropolis algorithm [69] is to

build a Markov chain where configurations are asymptotically generated according to P (xi), i.e.

lim

ti→∞

P (xi, ti) = Ptarget(xi) (2.69)

We can think of each step of the Markov chain at time ti as an operator π acting on the probability

Pti _{≡ P (x}

i, ti) [70]:

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We want a stationary state where P(t+1)_{= P}(t)_{: for i → ∞, we get}

Ptarget= Ptargetπ (2.71)

i.e Ptarget must be an eigenvector of π with eigenvalue equal to 1. It can be proved [70] that

all other eigenvectors have eigenvalues less than unity: thus, if the projection of the probability distribution at the first step P(1) _{on P}

target is not zero, P(1)πi will exponentially converge to the

target distribution for large ti. To determine π starting from Ptarget, it is easier to work with

conditional probabilities, since we effectively generate one configuration starting from the previous one. In the Metropolis algorithm, they are written as

PC(xA, ti|xB, tj) = T (xA|xB)A(xA|xB) (2.72)

This corresponds to a two-step process:

• given the state xj, a new configuration x0 is proposed according to an a priori transition

probability T (x0|xj)

• a test is performed with probability of success A(xi|xj). If passed, xi = x0; otherwise the

move is rejected and the old configuration counted one more time.

Since T (xi|x) must be a normalized probability,PiT (xi|x) = 1. We can rewrite eq. 2.70 using eq.

2.72: Pi+1(xA) = X xB T (xA|xB)A(xA|xB)Pi(xB) + T (xB|xA) 1 − A(xB|xA)Pi(xA) (2.73) i.e. the probability of being in xAat step i + 1 is the sum of the probabilities of accepting incoming

moves from other configurations (first term) and of rejecting moves leaving the configuration (second term). For the target probability we have that

Ptarget(xA) = X xB T (xA|xB)A(xA|xB)Ptarget(xB) + T (xB|xA) 1 − A(xB|xA)Ptarget(xA) (2.74) A sufficient (but not necessary) condition to satisfy eq. 2.74 is

PC(xB|xA)Ptarget(xA) = PC(xA|xB)Ptarget(xB) (2.75)

also known as the detailed balance condition, or microscopic reversibility. For a system of N particles, a simple implementation of the Metropolis algorithm is the following:

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• generate a random displacement vector ∆ whose components are uniformly distributed between −∆maxand ∆max

• propose a new state where the position of the selected particle is moved by ∆

In this way, the a priori transition probability is symmetric: T (xA|xB) = T (xB|xA). Eq. 2.75

becomes A(xB|xA) A(xA|xB) =Ptarget(xB) Ptarget(xA) (2.76)

A simple form of the acceptance probability A(xB|xA) that satisfies eq. 2.76 is A(xB|xA) =

minh1,Ptarget(xB)

Ptarget(xA)

i

2.3.2 Variational Monte Carlo

We now show how Born-Oppenheimer electronic energies can be written in a form satisfying eq. 2.64, using a method known as Variational Monte Carlo (VMC) [71–73]. Given a many body wavefunction ΨT(r1, . . . , rNe; {Rα}) ≡ ΨT(r; R) withR dr|ΨT(r; R)|

2_{= 1, the electronic energy in}

the Born-Oppenheimer approximation can be computed as

ET(R) = hΨT(r; R)| ˆH|ΨT(r; R)i (2.77)

In the position basis, eq. 2.77 can be rewritten as

ET(R) = Z drΨ∗_T(r; R) ˆH(r, R)ΨT(r; R) = Z dr|ΨT(r; R)|2EL(r; R) (2.78) EL(r; R) = ˆ H(r, R)ΨT(r; R) ΨT(r; R) (2.79)

|ΨT(r; R)|2≥ 0 in eq. 2.78 can be thought as a probability density and the integral as an average

of the quantity EL(r; R). Notice that if ΨT(r; R) is an eigenfunction of ˆH, EL(r; R) = ET(R) is

constant over all the electronic configurational space. An interesting observable is

σ2_T(R) = Z

dR|ΨT(r; R)|2 EL(r; R) − ET(R)2 =

Z

dR|ΨT(r; R)|2EL(r; R)2− ET(R)2(2.80)

that quantifies the fluctuations of the integrand around the average value. Again, if ΨT(r; R) is an

eigenfuction of ˆH, σ2 T = 0.

In our search for the ground state, we recall the variational principle:

E0(R) ≤ ET(R) for any ΨT(r; R) (2.81)

and the equality holds if ΨT(r; R) is the ground state wavefunction. ΨT(r; R) must satisfy some

conditions:

• ΨT(r; R) and ∇ΨT(r; R) must be continuous when the potential is finite

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• R dR|ΨT(r; R)|2EL(r; R)2must be finite as well, if we want a well defined σ2T

The practical problem is now evaluating ET(R) . This is achieved by employing the Monte Carlo

methods described in the previous section: ET(R) can be evaluated averaging EL(r; R) over a

3Nedimensional space. We must sample Nc configurations ri according to the probability density

|ΨT(r; R)|2 and then evaluate

ET(R) ≈ 1 Nc Nc X i=1 EL(ri; R) (2.82)

The configurations ri are generated through the Metropolis Monte Carlo algorithm by building the

corresponding Markov chain.

2.3.3 Our trial wavefunction

The key ingredient of the VMC calculation is the choice of the many-body trial wave function. The simplest antisymmetric wavefunction for a fermionic system with N↑spin-up and N↓spin-down electrons is a product of two Slater determinants of single electron orbitals:

For non-magnetic systems, N↑ = N↓. The wavefunction in eq. 2.83 can account for exchange effects, which keep like-spin electrons away from each other. Nevertheless, it can be an eigenfunction of a system of non interacting electrons only, where Coulomb repulsion is neglected. Moreover, there are some analytical constraints that a many-body wavefunction must satisfy that are not reproduced by single Slater determinants.

2.3.3.1 The Kato cusp conditions

One of such constraints is the so called Kato cusp condition [73–75]. This condition arises from the divergence in the Coulomb potential when the distance between two electrons becomes very small. If we explicitly consider electrons ~ri and ~rj and rewrite the Hamiltonian using the variables

~r = ~ri− ~rj and ~rcm=1₂(~ri+ ~rj), we obtain: ˆ H = −∇2 ~ r+ 1 r− 1 4∇ 2 ~rcm− 1 2 X k6=i,j ∇2 ~rk+ V (~r1, . . . , ~rN) − 1 r (2.85)

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where the first two terms are potentially divergent when ~r → 0 and the last term cancels the divergence present in the many body potential. If we fix the remaining electronic coordinates and expand the wavefunction in spherical harmonics, we obtain

Ψ(~r) ≡ ψ(~r; ~r1, . . . , ~rcm, . . . , ~rN) = X l l X m=−l rlflm(r)Ylm(θ, φ) (2.86)

where r,θ and φ are the spherical polar coordinates of ~r, Ylm the spherical harmonics and flm(r)

the coefficients of the expansion. The spin dependence is buried in these coefficients: if the two spins are parallel, the spatial part of the wavefunction must be odd in ~r and only flm(r) coefficients

with odd l survive; if the spins are antiparallel, the opposite is true and we only have non zero flm(r) coefficients when l is even. The contribution to the local energy coming from the first two

terms in eq. 2.85 for antiparallel and parallel spins for small ~r is

E_La = −∇ 2_Ψ(~_r) Ψ(~r) + 1 ~ r = − ∇2_f 00(r) f00(r) + O(r0) +1 r = − ∇2_f 00(r) f00(r) + O(r0) +1 r = = − 2 rf00(0) ∂f00 ∂r |r=0+ O(r 0_{) +}1 r (2.87) E_Ls = −∇ 2_Ψ(~_r) Ψ(~r) + 1 ~ r = − ∇2h_rP1 m=−1f1m(r)Y1m(θ, φ) i rP1 m=−1f1m(r)Y1m(θ, φ) + O(r0) +1 r = = −4 h P1 m=−1 ∂f1m ∂r |r=0Y1m(θ, φ) i rP1 m=−1f1m(0)Y1m(θ, φ) + O(r0) +1 r (2.88)

where only divergent terms of the expansions are retained, while the regular behaving terms are absorbed in the O(r0) term. If the wavefunction is an Hamiltonian eigenfunction, the local energy is a constant everywhere and the divergent terms in eq. 2.87 and 2.88 must cancel each other for every value of θ and φ, i.e.

∂f00 ∂r _r=0 = f00(0) 2 (2.89) ∂f1m ∂r _r=0 =f1m(0) 4 (2.90)

Since there is no explicit correlation, Slater determinants of single particle orbitals cannot depend on interelectronic coordinates, and cannot enforce the conditions described in eq. 2.89 and 2.90. An analogous cusp condition must be satisfied when electrons are in the proximity of a nucleus. With similar arguments, one can prove that an electron-nucleus cusp condition exists:

∂Ψ00 ∂ri _r i=0 = −ZΨ00 (2.91)

where ri is the distance of the i-th electron from a nucleus with charge Z and Ψ00 is the spherical

average of the wavefunction around the nucleus. This condition can either be satisfied by the Slater determinant (if and only if each individual orbital satisfies the cusp condition) or implemented by modifying the wavefunction.

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A popular form for the trial wavefunction is

ΨT(r|R) = eJ (r|R)S↑[θk(~ri|R)]S↓[θk(~ri|R)] (2.92)

where J (r|R) is the so called Jastrow factor [76]. Since the antisymmetric character of the wavefunction is built into the Slater determinant, the Jastrow factor must be symmetric under particle permutation. For the Jastrow-Slater wavefunction, eq. 2.89, 2.90 and 2.91 become

∂J ∂r _r=0 = 1 2 antiparallel spins (2.93) ∂J ∂r _r=0 = 1 4 parallel spins (2.94) ∂J ∂r _r=0 = −Z nuclei (2.95)

The last equality is valid if we assume that the Slater determinant is smooth with respect to electronic coordinates relative to nuclei (this is the case if, for example, a non divergent pseudopotential is used to compute the single orbitals). If, on the other hand, every electronic orbital satisfies the nuclear cusp condition, the determinant satisfies the same condition as well and the Jastrow factor must be cuspless (i.e. Z = 0). In our case, the Jastrow factor is written as

J (r|R) = − Ne X i=1   1 2 Ne X j6=i uee(rij) − Np X α=1 uep(|~ri− ~Rα|)   (2.96)

where rij = |~ri− ~rj|, satisfying the symmetric constraint required by the global fermionic

wave-function. An important indication on the form of the uee and uep comes from the Random Phase

Approximation [77]. In this approximation the Hamiltonian of the system is reduced to a sum of a short-range interactions among electrons and a long range part described by collective oscillations (plasmons). Since RPA becomes formally exact when the electron density goes to infinity, results obtained in this approximation can be useful for our high-density regime. In particular, it can be proved [77, 78] that minimizing the energy in RPA leads to

uRP A_ee (k) = −1 2 + √ 1 + ak uRP A_ep (k) = −√ ak 1 + ak (2.97)

where uee(k) and uep(k) are the Fourier transforms of uee(r) and uep(r), and ak= 12rs/k4, with

rs= 3v_4π

(1/3)

. The RPA forms satisfy the cusp conditions, providing the expected analytic limits both for r → 0 and r → ∞: still, the RPA does not provide the exact solution for intermediate values of r. Following ref. [79], we use:

˜ uα(r) = uRP Aα (r) + λα2b e−(r/w α 2b) 2 α = (ee, ep) (2.98)

adding a remaining empirical part to uRP Aα (r), which is a simple Gaussian preserving both short

and long range behavior from RPA and introducing the free variational parameters λα

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2.3.3.2 Backflow transformation

The inclusion of the Jastrow factor provides a fraction of the electronic correlation missing in a simple Slater determinant. However, the form of the wavefunction can still be improved. A formal expression of the exact ground state wavefunction in terms of a trial wavefunction is the Feynman-Kac formula [55]: Ψ0(r|R) = ΨT(r|R) hΨT|Ψ0i hexp − Z ∞ 0 dt (EL(rt) − E0(R)) iψ2 T (2.99)

where Ψ0(r|R) is the exact many-body wavefunction; ΨT(r|R) is the (real, non negative) trial

wavefunction; E0is the exact ground energy; h. . . iΨ2

T stands for an average over different trajectories

all starting from r and evolving for a time t according to drt

dt = η(t) − ∇ ln ψT(rt|R) (2.100)

η(t) is a Weiner process [68], i.e. the electrons perform a Brownian motion, being at the same time under the effect of the drift −∇ ln ψT(rt|R). The above expression can be approximated [80]: if the

trial wavefunction is good enough, the exponent will be small and the cumulant approximation can be invoked, i.e. Ψ0(r|R) ∝ ΨT(r|R)hexp − Z ∞ 0 dt(EL(rt) − E0(R))iΨ2 T ≈ ≈ ΨT(r|R) exp −h ¯E − E0(R)iΨ2 T + 1 2h ¯δE 2_i Ψ2 T (2.101) where ¯E = R∞

0 dtEL(rt). If the expansion is truncated at the first term and some simplifying

ansatzs are assumed, one finds that the electronic coordinates in the Slater determinant part of the wavefunction are replaced by the backflow coordinates xi,

~xi= ~ri+ Ne X j6=i ˜yRP A_ee (rij) (~ri− ~rj) + Np X α=1 h ˜ yRP A_ep (|~ri− ~Rα|)) i (~ri− ~Rα) (2.102)

where the analytical form of ˜yee and ˜yepcan be found in ref. [80]. Empirical variational parameters

can be added on top of these analytical expressions to make the trial wavefunction more flexible. Gaussians were first introduced by Kwon et al. [81] for the homogeneous electron gas; they were later used for high-pressure hydrogen [82], improving both the variational energy and the associated variance. The ˜yRP Aee and ˜yRP Aep are replaced by

˜ yα = yαRP A(rα) + ηep(rα) ηα(r) = λαb e−((r−r α b)/w α b) 2 α = (ee, ep) (2.103)

with free variational parameters λα_b, r_bαand wα_b. The final explicit form of the trial wave function is